Background.

    Consider the second order linear differential equation
    
(1)        .

Rewrite this equation in the form   ,  then use the substitutions    and    and rewrite the differential equation (1) in the form  

(2)        .

Definition (Analytic).  The functions and are analytic at if they have Taylor series expansions with radius of convergence and  , respectively.  That is  

          which converges for  
    and
          which converges for  

Definition (Ordinary Point).  If the functions and are analytic at , then the point is called an ordinary point of the differential equation  

        .

Otherwise, the point is called a singular point.  

Definition (Regular Singular Point).  Assume that is a singular point of (1) and that   and   are analytic at .

They will have Maclaurin series expansions with radius of convergence and  , respectively.  That is  

          which converges for  
    and
          which converges for  

Then the point   is called a regular singular point of the differential equation (1).

Method of Frobenius.

    This method is attributed to the german mathemematican Ferdinand Georg Frobenius (1849-1917 ).  Assume that   is regular singular point of the differential equation

    .  

    
A Frobenius series (generalized Laurent series) of the form  

          

can be used to solve the differential equation.  The parameter must be chosen so that when the series is substituted into the D.E. the coefficient of the smallest power of   is zero.  This is called the indicial equation.  Next, a recursive equation for the coefficients is obtained by setting the coefficient of    equal to zero.  Caveat: There are some instances when only one Frobenius solution can be constructed.

Proof  Frobenius Series Method  Frobenius Series Method  

Definition (Indicial Equation).  The parameter in the Frobenius series is a root of the indicial equation

        .

Assuming that the singular point is  , we can calculate as follows:

        
and
        
Derivation.

The Recursive Formulas.  

    For each root of the indicial equation, recursive formulas are used to calculate the unknown coefficients .  This is custom work because a numerical value for is easier use.  

Example 1.  Use Frobenius series to solve the D. E.   
        .  
Solution 1.

Example 2.  Use Frobenius series to solve the D. E.   
        .  
Solution 2.

Example 3.  Use Frobenius series to solve the D. E.   
        .  
Solution 3.

Example 4.  Use Frobenius series to solve the D. E.   
        .  
A solution is known to be the celebrated Bessel function  .
Solution 4.

Example 5.  Use Frobenius series to solve the D. E.   
        .  
A solution is known to be the celebrated Bessel function  .
Solution 5.

Example 6.  Use Maclaurin series and verify the identity  .
Solution 6.

Application of the Vibrating Drum

    The two dimensional wave equation is   ,  

in rectangular coordinates it is   ,  

and in polar coordinates it is   .

     Consider a drum head that a flexible circular membrane of radius .  Assume that it is struck in the center and this produces radial vibrations only where the displacement depends only on time and distance from the center.  Then   satisfies the D.E.
      
    .

Proof Vibrating Drum  Vibrating Drum  

Example 7.  Consider a drum head of radius . For convenience, choose the parameter  . The method of separation of variables permits us to use the substitution  .  Use this substitution and obtain the D.E.
    .      
Solve this D.E. and plot the solution over the interval  .
Solution 7.

Example 8.  In Example 7, the boundary condition for the D.E. is  ,  i.e. the drum head has radius  .
Thus the parameter    must be chosen to be a root of the Bessel function.
The zeros do not have a simple formula. However it is known that they are "close to" multiples of  .  
Verify this and find the first five zeros.
Solution 8.

Surface equation for the vibrating drum.

    The solution we are seeking in Example 7 is   where the boundary condition   requires that  ,  hence  . Therefore the fundamental solutions to the wave equation for the drum head is  

    ,  for  n = 1,2,3.  

Example 9.  Plot the functions   is the n-th root of  .
Since we will be considering a drum of unit radius, plot   over the interval  .
Solution 9.

Example 10.  The initial displacement for a fundamental solution is  .  
Plot the functions for  n = 1,2,3.  
The first fundamental solution vibrates up and down throughout the entire disk of radius 1.
Solution 10.

Old Lab Project (Frobenius Series Solution of O.D.E.'s  Frobenius Series Solution of O.D.E.'s).  Internet hyperlinks to an old lab project.  

Old Lab Project (Bessel Functions and Vibrating Drum  Bessel Functions and Vibrating Drum).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Series Solutions and Frobenius Method  Series Solutions and Frobenius Method Internet hyperlinks to web sites and a bibliography of articles.  

Vibrating Drum  Vibrating Drum  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook  Frobenius Series Solution

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(c) John H. Mathews 2004